Solve following: frac{cos 45^{circ}}{sec 30^{circ}+operatorname{cosec} 30^{circ}}

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8. Introduction to Trigonometry

hard

Evaluate the following:

$\frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$

Option A

Option B

Option C

Option D

Solution

$\frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$

$=\frac{\frac{1}{\sqrt{2}}}{\frac{2}{\sqrt{3}}+2}=\frac{\frac{1}{\sqrt{2}}}{\frac{2+2 \sqrt{3}}{\sqrt{3}}}$

$=\frac{\sqrt{3}}{\sqrt{2}(2+2 \sqrt{3})}=\frac{\sqrt{3}}{2 \sqrt{2}+2 \sqrt{6}}$

$=\frac{\sqrt{3}(2 \sqrt{6}-2 \sqrt{2})}{(2 \sqrt{6}+2 \sqrt{2})(2 \sqrt{6}-2 \sqrt{2})}$

$=\frac{2 \sqrt{3}(\sqrt{6}-\sqrt{2})}{(2 \sqrt{6})^{2}-(2 \sqrt{2})^{2}}=\frac{2 \sqrt{3}(\sqrt{6}-\sqrt{2})}{24-8}=\frac{2 \sqrt{3}(\sqrt{6}-\sqrt{2})}{16}$

$=\frac{\sqrt{18}-\sqrt{6}}{8}=\frac{3 \sqrt{2}-\sqrt{6}}{8}$

Standard 10

Mathematics

If $\tan ( A + B )=\sqrt{3}$ and $\tan ( A – B )=\frac{1}{\sqrt{3}} ; 0^{\circ}< A + B \leq 90^{\circ} ; A > B ,$ find $A$ and $B$

medium

Evaluate:

$\sin 25^{\circ} \cos 65^{\circ}+\cos 25^{\circ} \sin 65^{\circ}$

medium

$\sin 2 A=2 \sin A$ is true when $A=$

easy

State whether the following are true or false. Justify your answer.

$(i)$ $\cos A$ is the abbreviation used for the cosecant of angle $A$

$(ii)$ cot $A$ is the product of cot and $A$.

$(iii)$ $\sin \theta=\frac{4}{3}$ for some angle $\theta$.

medium

Evaluate:

$\cos 48^{\circ}-\sin 42^{\circ}$

easy

Evaluate the following: $\frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$

Solution

Similar Questions

If $\tan ( A + B )=\sqrt{3}$ and $\tan ( A – B )=\frac{1}{\sqrt{3}} ; 0^{\circ}< A + B \leq 90^{\circ} ; A > B ,$ find $A$ and $B$

Evaluate: $\sin 25^{\circ} \cos 65^{\circ}+\cos 25^{\circ} \sin 65^{\circ}$

$\sin 2 A=2 \sin A$ is true when $A=$

State whether the following are true or false. Justify your answer. $(i)$ $\cos A$ is the abbreviation used for the cosecant of angle $A$ $(ii)$ cot $A$ is the product of cot and $A$. $(iii)$ $\sin \theta=\frac{4}{3}$ for some angle $\theta$.

Evaluate: $\cos 48^{\circ}-\sin 42^{\circ}$

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Evaluate the following:

$\frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$

Evaluate:

$\sin 25^{\circ} \cos 65^{\circ}+\cos 25^{\circ} \sin 65^{\circ}$

State whether the following are true or false. Justify your answer.

$(i)$ $\cos A$ is the abbreviation used for the cosecant of angle $A$

$(ii)$ cot $A$ is the product of cot and $A$.

$(iii)$ $\sin \theta=\frac{4}{3}$ for some angle $\theta$.

Evaluate:

$\cos 48^{\circ}-\sin 42^{\circ}$